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12   QUOTIENT GROUPS       57

There is also a canonical bijection if/ of the quotient group A/(A n H) onto
the subgroup n(A) of G/H; this bijection \j/ is the unique mapping which makes
the following diagram commutative:


(12.12.4)                                ,/

A/(AnH)-^ 7t(A)

where n' is the canonical mapping and 7 : A ->G the canonical injection. It
follows from (12.10.6) that \// is continuous, but it is not necessarily bicon-
tinuous (Problem 2). However:

(12.12.5)    Let G be a metrizable group and H a compact subgroup ofG. For
each closed subgroup A of G, the canonical bijection

\//: A/(A n H) -> 7t(A)
is an isomorphism of topological groups.

It is enough to show that the image under \f/ of an arbitrary closed subset F
of A/(A n H) is closed in 7i(A). Now ^"'(F) = B is a closed subset of A,
hence closed in G, and ^(F) = n(j(B)) = 7i(BH). But since H is compact, the
set BH is closed in G (12.10.5) and saturated, hence 7r(BH) is closed in G/H
(12.10) and therefore also closed in 7r(A).

(12.12.6)    Let Gj, G2 be two topological groups and Hi (resp. H2) a normal
subgroup ofGl (resp. G2). Then the canonical bijection

, x H2) -Kd/H,) x (G2/H2)
is an isomorphism of topological groups.

This is an immediate consequence of (12.10.11).

(12.12.7) Let G, G' be two topological groups and u: G ~G' a continuous
homorphism. Then u factorizes canonically as

G 4> G/N 4- M(G) -i G'

where N is the kernel of w,/? is the canonical homomorphism, and j the canon-
ical injection. The mapping v is a continuous bijection (12.10.6(iii)), but is not
necessarily bicontinuous. When v is bicontinuous, the homomorphism u is said
to be a .yfnttf morphism of G into G'; for this to be the case, it is necessary and
sufficient that for each neighborhood V of e in G, the set u(V) should be a
neighborhood of e' = u(e) in u(G) (12.11.5).plied to the right action